Outside Good Uncertainty

\[ \newcommand{\x}{\mathbf{x}} \newcommand{\bfbeta}{\boldsymbol{\beta}} \newcommand{\iid}{\stackrel{iid}{\sim}} \]

1 Introduction

Motivate with:

1. Anecdotes:

   • Quote stats on cart abandonment rates. Consumers can pick a preferred bluetooth speaker out of many brands (or a preferred color out of many options from the same brand), but then they may hesitate making the purchase.
   • no unusual reaction from consumers if you ask them “how likely are you to purchase?” Uncertainty about making a purchase “feels natural.”

2. Psych findings:

   • [possibly use] decoy effect: low purchase prob when coke machine by iteslf, but high coke and pepsi purchase prob when machines side-by-side,

3. Practice:

   • Brazell et al 2006 paper
   • Sawtooth’s implementation in Lighthouse Studio application

Describe: The dual-response framework asks for an immediate comparison (which of these things in front of you do you like best) and a hypothetical future scenario (if this product was available to purchase in the near future, would you purchase it). We provide a behavioral model that rationalizes use of the dual-response framework

2 Model

2.1 Set Up

Consumer \(i=1,\ldots,N\) derives utility from good \(j \in \mathcal{J}_i = \lbrace 0,1,2,\ldots,J_i \rbrace\) with utility \(u_{ij}\) given by \[ u_{ij} = h(\x_j, \bfbeta_i) + \eta_{ij}. \] where \(\x_j\) is a length-\(P\) column vector of good characteristics, \(\bfbeta_i\) is a length-\(P\) column vector of consumer-specific taste parameters, and \(\eta_{ij}\) encapsulates factors known to the consumer but unobserved by the researcher. We take \(h(\x_j, \bfbeta_i) = \x_j'\bfbeta_i\) but this specification is not required.

The “zero”-th (or “outside”) good is special and is associated with a zero vector of good characteristics (\(\x_0 = \mathbf{0}\)) such that \(h(\x_0, \bfbeta_i)=0\). Consumers observe \(\x_j\) and \(\eta_{ij}\) for all “inside” goods (\(j \in \mathcal{J}^{+}_i \coloneqq \lbrace 1,2,\ldots,J_i \rbrace\)), but they do not observe \(\eta_{i0}\).

The consumer first reports her preferred inside good, which is given by \[ j^*_i = \underset{j \in \mathcal{J}^+}{\operatorname{arg max}} \ u_{ij}. \]

The consumer then reports a value \(y_i\) on a discrete qualitative scale \(w \in \mathcal{W} = \lbrace 1, \ldots, W \rbrace\) to reflect the probability \(p_i\) that she would purchase \(j^*\), i.e., that she prefers good \(j^*_i\) to the outside good \(j=0\).

Let \(F\) denote the cumulative distribution function associated with \(p_i\). Apportion the range of \(F\) into a number \(W\) of intervals \(\phi_w = \left[ \alpha_{w-1}, \alpha_w \right)\), where \(\alpha_0 = 0\) and \(\alpha_W = 1\). We assume that consumers share the definition of the qualitative scale and thus understand these intervals, whereas \(\alpha_w\) for \(0<w<W\) are unobserved by the researcher and are to be estimated.

Let \(u_i^*\) denote the utility of good \(j^*_i\) for consumer \(i\). Consumers know \(u_i^*\), but do not know \(\eta_{i0}\), and thus the purchase probability from the consumer’s perspective is given by \[ p_i = \Pr \left( \eta_{i0} < u_i^* \right) = F\left( u_i^* \right). \]

The act of reporting the interval \(y_i=w\) into which \(p_i\) falls, reveals that \(p_i \in \phi_{w\left(i\right)} = \left[\alpha_{w-1},\alpha_w\right)\) and is therefore equivalent to reporting that \[ u_i^* \in \left[ F^{-1} \left( \alpha_{w_{i-1}} \right), \; F^{-1} \left( \alpha_{w_i} \right) \right]. \]

2.2 Probability Specification

We model \(\eta_{ij} \iid \text{Gumbel} \left( 0,1 \right)\) with \(\eta_{ij}\) assumed to be independent of both \(\x_j\) and \(\eta_{ik}\) for \(k \ne j\),^{1 2}

From the consumer’s perspective, her purchase probability is \[ p_i = \Pr \left( \eta_{i0} < u_i^* \right) = F_{\text{Gumble}\left(0,1\right)} \left( u_i^* \right) \] and, before she reports \(y_i\), the probability of reporting any particular value \(w\) on the ordinal scale \(\Pr(y_i=w)\) is given by \[ \begin{aligned} \Pr(y_i=w) &= F_{\text{Gumbel}\left(0,1\right)} \left( F^{-1}_{\text{Gumbel}\left(0,1\right)} \left( \alpha_w \right) \right) - F_{\text{Gumbel}\left(0,1\right)} \left( F^{-1}_{\text{Gumbel}\left(0,1\right)} \left( \alpha_{w-1} \right) \right) \\ &= \alpha_w - \alpha_{w-1}. \end{aligned} \]

From the researcher’s perspective, the unobserved likelihood of purchase is \[ p_i = \Pr \left( \eta_{i0} - u_i^* < 0 \right) = \Pr \left( \varepsilon_i < 0 \right) \] where \(\varepsilon_i = \eta_{i0} - u_i^*\). As is well known,³ \(u_i^*\) follows a Gumbel distribution with location parameter \(\overline{\mu}_i\) and scale parameter \(1\), where \[ \overline{\mu}_i = \ln \left( \sum_{j\in\mathcal{J}_i} \exp \left( {\x_j}'\bfbeta_i \right) \right). \tag{1}\] and so \(\varepsilon_i\) follows a Logistic distribution with location parameter \(\overline{\mu}_i\) and scale parameter \(1\). The unobserved choice probability is \[ p_i = F_{\text{Logistic} \left( \overline{\mu}_i, 1 \right)} \left( \varepsilon_i \right). \]

Therefore, the probability that consumer \(i\) reports \(y_i=w\) is \[ \begin{aligned} \Pr \left( u_i^* \hspace{1ex} \text{s.t.} \hspace{1ex} p_i \in \phi_{w\left(i\right)} \right) &= F_{\text{Logistic} \left( \overline{\mu}, 1 \right) } \left( F^{-1}_{\text{Gumbel} \left( 0, 1 \right) } \left( \alpha_{w\left(i\right)} \right) \right) - F_{\text{Logistic} \left( \overline{\mu}, 1 \right) } \left( F^{-1}_{\text{Gumbel} \left( 0, 1 \right) } \left( \alpha_{w\left(i\right)-1} \right) \right) \\ &= \frac{1}{1 - \exp \left( \overline{\mu} \right) \ln \alpha_{w\left(i\right)}} - \frac{1}{1 - \exp \left( \overline{\mu} \right) \ln \alpha_{w\left(i-1\right)}} \end{aligned} \]

3 Simulation Study

1. Agg MNL

• Simulate data (\(\bfbeta_i = \bfbeta\)), then:
• recover parameters
• compare to dichotomized dual-response
• compare to constant-prob (if \(W=5\) set \(\alpha\)’s such that \(\Pr(y_i=w) = 1/5\) )
• or, instead of 1/W do the observed frequency

1. HB MNL

• Same 4 things as above

1. Assessing \(\alpha\) cut-points

• May need to try extreme values for some \(\alpha\)’s
• What happens when different consumers have different interpretation of the \(W\) categories such that \(\alpha\)’s are not the same for all consumers?

4 Empirical Analysis

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5 Discussion

Differs from (Brazell et al. 2006)

6 Conclusion

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References

Brazell, Jeff D., Christopher G. Diener, Ekaterina Karniouchina, William L. Moore, Válerie Séverin, and Pierre-Francois Uldry. 2006. “The No-Choice Option and Dual Response Choice Designs.” Marketing Letters 17 (4): 255–68. https://doi.org/10.1007/s11002-006-7943-8.

Cardell, N. Scott. 1997. “Variance Components Structures for the Extreme-Value and Logistic Distributions with Application to Models of Heterogeneity.” Econometric Theory 13 (2): 185–213. https://doi.org/10.1017/s0266466600005727.

McFadden, Daniel L. 1981. “Structural Discrete Probability Models Derived from Theories of Choice.” In, 198–272. The MIT Press.

Footnotes

1. This can be motivated by a framework in which \(u_0 = 0\) and \(u_{ij} = {\x_j}'\bfbeta_i + \eta_{ij} - \eta_0\) for \(j \in \mathcal{J}^{+}_i\). Here, \(\eta_0\) captures the consumer’s uncertainty about their future tastes. Given that utilities are ordinal and \(\eta_0\) is a common shock, it plays no role in the choice among the most preferred inside good \(j^* \in \mathcal{J}^{+}_i\).

2. We note that, while the parametrization of \(\eta_0 \sim \text{Gumbel}\left(0,1\right)\) preserves symmetry among the \(J+1\) goods and is thus a natural choice, the framework can easily accommodate an alternative distribution for \(\eta_0\). For example, one could use an affine function of individual characteristics to accommodate individual-level variation in the propensity to prefer the outside good.

3. See, e.g., (McFadden 1981) or (Cardell 1997).